Efficient evaluation of multivariate polynomials over multidimensional grids

Let P(x
1
, x
2
, ... , x
n
) be a multivariate polynomial in n variables, x
1
, x
2
, ... , x
n
and suppose that P(x
1
, x
2
, ... , x
n
) is of degree m
i
in x
i
, i = 1, 2, ... , n. It is shown how P(x
1
, x
2
, ... , x
n
) may be evaluated at x
i
= x
0i
+ n
i
Δ
i
for n
i
= 0, 1, 2, ... and i = 1, 2, ... , n using not more than (m
1
+ m
2
+ ... + m
n
) additions per point. Feasibility of reduction from this upper bound is demonstrated in the n = 2 case.